Are liberals more unhappy than conservatives?

In short, yes.

Over the past 40 years, when asked by the General Social Survey, “Taken all together, how happy would you say you are these days? Would you say that you are very happy, pretty happy, or not too happy?”, self-identified conservatives in general reported feeling "very happy" at rates higher than their liberal counterparts.

Data source: General Social Survey, 1974 to 2004.
Image source: http://freakonomics.com/2008/04/23/conservatives-are-happier-than-liberals-discuss/

However, correlation does not imply causation. There are a number of factors that are even bigger predictors for happiness and in fact, wealth is one of the biggest. This makes sense because with juggling finances and other money-related concerns, there is less time for making yourself happy or taking time for leisure. Any person who has taken an introductory psychology course can probably also tell you this as well because they were probably introduced to Maslow's hierarchy of needs which says that it's pretty hard to focus on bringing esteem into your world when you're focusing on simply maintaining your own physical needs. It also turns out that the richer you are, the more likely you're going to vote republican (probably indicating that you're a conservative). 

Rush Limbaugh tries to make the case that the liberal worldview is one of "pessimism, darkness, and sadness" and then quite shockingly ties this into the suicide of Robin Williams. Rush's lack of decency aside, even if Rush did have access to this data, one would be hard-pressed to prove a causal relationship between the "liberal worldview" and one's happiness or mental health.

However, there is evidence that the conservative worldview or something like it does promote happiness, specifically the belief that the best outcome for society is achieved when individuals work towards their own goals. In a study analyzing how happy subjects were based on whether they performed "self-interested" actions, it was found that participants are much more happy when they do things for their own interests. In the context of politics, Eric Horowitz points out:

Conservatives are more likely than liberals to believe that strong societies are made when people strive for individual greatness rather than concern themselves with collective well-being — i.e. the best way to increase society’s “sum of its parts” is for each part to focus on maximizing its own value. The notion that acting selfish will improve yourself and your society essentially “imposes” self-interested actions on people. If you’re a good person who cares about others you have no choice but to act in a self-interested manner. Because liberals are less taken with this view, they will perceive a real choice between self-interested and prosocial actions, and thus when they do engage in self-interested actions they will derive less happiness from them. 

As much as I hate to admit it then, Rush might be right about the liberal worldview to an extent. But is liberal unhappiness really a bad thing? If liberals aren't as happy when they serve their own needs, then maybe they're more receptive to the growing income inequality in this country, the imminent threat of global warming and the next mass extinction, or the fact that minorities are disproportionately affected by our criminal justice system. If that's the case, then maybe these are good things to be unhappy about.